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  Leetcode-118-杨辉三角
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      <h1 id="Leecode-118-Pascal’s-Triangle"><a href="#Leecode-118-Pascal’s-Triangle" class="headerlink" title="Leecode-118-Pascal’s Triangle"></a>Leecode-118-<a href="https://leetcode-cn.com/problems/pascals-triangle/" target="_blank" rel="noopener">Pascal’s Triangle</a></h1><h2 id="思路：动态规划"><a href="#思路：动态规划" class="headerlink" title="思路：动态规划"></a>思路：动态规划</h2><h2 id="题目描述："><a href="#题目描述：" class="headerlink" title="题目描述："></a>题目描述：</h2><p><img src="http://zhuuu-bucket.oss-cn-beijing.aliyuncs.com/img/20200501/114008082.png" alt="mark"></p>
<p>给定一个负整数numRows，生成杨辉三角的前numRows行。（在杨辉三角中，每个数是它左上方和右上方的数的和。）</p>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br></pre></td><td class="code"><pre><span class="line">输入: 5</span><br><span class="line">输出: 杨辉三角的前五行</span><br><span class="line">[</span><br><span class="line">     [1],</span><br><span class="line">    [1,1],</span><br><span class="line">   [1,2,1],</span><br><span class="line">  [1,3,3,1],</span><br><span class="line"> [1,4,6,4,1]</span><br><span class="line">]</span><br></pre></td></tr></table></figure>



<a id="more"></a>



<p><strong>Solution：动态规划</strong></p>
<ul>
<li><p>思路：如果能知道上一行的杨辉三角，就可以根据每对相邻的值轻松计算出下一行。</p>
</li>
<li><p>首先：我们先生成整个<code>triange</code>矩阵，三角形的每一行都以子列表的形式存储。</p>
</li>
<li><p>然后，检查初始化情况，如果输入的行数为0,就会返回[1]。（也就是杨辉三角的第一行）</p>
</li>
<li><p>最后，如果numRows（输入的行数）大于0的话，那么我们用[1] 来初始化第一行。</p>
</li>
</ul>
<p>具体填充方式如下所示：</p>
<p><img src="http://zhuuu-bucket.oss-cn-beijing.aliyuncs.com/img/20200501/115050454.png" alt="mark"></p>
<p><img src="http://zhuuu-bucket.oss-cn-beijing.aliyuncs.com/img/20200501/115102275.png" alt="mark"></p>
<p><img src="http://zhuuu-bucket.oss-cn-beijing.aliyuncs.com/img/20200501/115113828.png" alt="mark"></p>
<p><img src="http://zhuuu-bucket.oss-cn-beijing.aliyuncs.com/img/20200501/115119450.png" alt="mark"></p>
<h2 id="Java"><a href="#Java" class="headerlink" title="Java"></a>Java</h2><p><strong>Solution :</strong></p>
<figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br></pre></td><td class="code"><pre><span class="line"><span class="class"><span class="keyword">class</span> <span class="title">Solution</span></span>&#123;</span><br><span class="line">    <span class="keyword">public</span> List&lt;List&lt;Integer&gt;&gt; generate(<span class="keyword">int</span> numRows)&#123;</span><br><span class="line">        <span class="comment">// 返回结果的数组</span></span><br><span class="line">        List&lt;List&lt;Integer&gt;&gt; triangle = <span class="keyword">new</span> ArrayList&lt;&gt;();</span><br><span class="line"></span><br><span class="line">        <span class="comment">// 第一步：特殊判断</span></span><br><span class="line">        <span class="keyword">if</span> (numRows == <span class="number">0</span>)&#123;</span><br><span class="line">            <span class="keyword">return</span> triangle;</span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        <span class="comment">// 第二步：初始化数组 把第一行变成[[1]]</span></span><br><span class="line">        triangle.add(<span class="keyword">new</span> ArrayList&lt;&gt;());</span><br><span class="line">        triangle.get(<span class="number">0</span>).add(<span class="number">1</span>);</span><br><span class="line"></span><br><span class="line">        <span class="comment">// 第三步：状态转移(从第二行开始)</span></span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> rowNum = <span class="number">1</span>; rowNum &lt; numRows; rowNum++) &#123;</span><br><span class="line">            <span class="comment">// 当前行的元素</span></span><br><span class="line">            List&lt;Integer&gt; row = <span class="keyword">new</span> ArrayList&lt;&gt;();</span><br><span class="line">            <span class="comment">// 上一行的元素</span></span><br><span class="line">            List&lt;Integer&gt; preRow = triangle.get(rowNum - <span class="number">1</span>);</span><br><span class="line"></span><br><span class="line">            <span class="comment">// 第一个元素永远是1</span></span><br><span class="line">            row.add(<span class="number">1</span>);</span><br><span class="line"></span><br><span class="line">            <span class="comment">// 中间的每个元素是上一行的 下标j - 1元素 加上 j下标的元素</span></span><br><span class="line">            <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">1</span>;j &lt; rowNum; j++)&#123;</span><br><span class="line">                row.add(preRow.get(j-<span class="number">1</span>) + preRow.get(j));</span><br><span class="line">            &#125;</span><br><span class="line"></span><br><span class="line">            <span class="comment">// 最后一个元素永远是1</span></span><br><span class="line">            row.add(<span class="number">1</span>);</span><br><span class="line"></span><br><span class="line">            <span class="comment">// 将每次的结果加到triangle的结果中</span></span><br><span class="line">            triangle.add(row);</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">return</span> triangle;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>



<p><strong>测试用例：</strong></p>
<figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">MainClass</span> </span>&#123;</span><br><span class="line">    <span class="comment">// Integer 数组转字符串输出</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> String <span class="title">integerArrayListToString</span><span class="params">(List&lt;Integer&gt; nums, <span class="keyword">int</span> length)</span></span>&#123;</span><br><span class="line">        <span class="keyword">if</span> (length == <span class="number">0</span>)&#123;</span><br><span class="line">            <span class="keyword">return</span> <span class="string">"[]"</span>;</span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        String result = <span class="string">""</span>;</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> index = <span class="number">0</span>;index &lt; length;index++)&#123;</span><br><span class="line">            Integer num = nums.get(index);</span><br><span class="line">            result += Integer.toString(num) + <span class="string">", "</span>;</span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        <span class="keyword">return</span> <span class="string">"["</span> + result.substring(<span class="number">0</span>,result.length() - <span class="number">2</span>) + <span class="string">"]"</span>;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="comment">// Integer 数组转字符串输出 重载</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> String <span class="title">integerArrayListToString</span><span class="params">(List&lt;Integer&gt; nums)</span></span>&#123;</span><br><span class="line">        <span class="keyword">return</span> integerArrayListToString(nums,nums.size());</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="comment">// 二维List&lt;List&lt;Integer&gt;&gt; 转字符串</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> String <span class="title">int2dListToString</span><span class="params">(List&lt;List&lt;Integer&gt;&gt; nums)</span></span>&#123;</span><br><span class="line">        StringBuilder sb = <span class="keyword">new</span> StringBuilder(<span class="string">"["</span>);</span><br><span class="line">        <span class="keyword">for</span>(List&lt;Integer&gt; list:nums)&#123;</span><br><span class="line">            sb.append(integerArrayListToString(list));</span><br><span class="line">            sb.append(<span class="string">","</span>);</span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        sb.setCharAt(sb.length() - <span class="number">1</span>,<span class="string">']'</span>);</span><br><span class="line">        <span class="keyword">return</span> sb.toString();</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">main</span><span class="params">(String[] args)</span> <span class="keyword">throws</span> IOException </span>&#123;</span><br><span class="line">        BufferedReader in = <span class="keyword">new</span> BufferedReader(<span class="keyword">new</span> InputStreamReader(System.in));</span><br><span class="line">        String line;</span><br><span class="line">        <span class="keyword">while</span> ((line = in.readLine()) != <span class="keyword">null</span>)&#123;</span><br><span class="line">            <span class="keyword">int</span> numRows = Integer.parseInt(line);</span><br><span class="line">            List&lt;List&lt;Integer&gt;&gt; ret = <span class="keyword">new</span> Solution().generate(numRows);</span><br><span class="line">            String out = int2dListToString(ret);</span><br><span class="line">            System.out.println(out);</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>



<ul>
<li>时间复杂度：<code>O(numRows^2)</code> ，两层循环，每层要运行<code>rowNum</code>次</li>
<li>空间复杂度：<code>O(numRows^2)</code>，额外需要的dp数组大小</li>
</ul>
<p><img src="http://zhuuu-bucket.oss-cn-beijing.aliyuncs.com/img/20200501/123811179.png" alt="mark"></p>
<h2 id="不讲武德"><a href="#不讲武德" class="headerlink" title="不讲武德"></a>不讲武德</h2><figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment">// 重点：ArrayList如何初始化二维数组</span></span><br><span class="line"><span class="class"><span class="keyword">class</span> <span class="title">Solution</span> </span>&#123;</span><br><span class="line">    <span class="keyword">public</span> List&lt;List&lt;Integer&gt;&gt; generate(<span class="keyword">int</span> numRows) &#123;</span><br><span class="line">        Integer[][] a= &#123;&#123;<span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">2</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">3</span>, <span class="number">3</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">4</span>, <span class="number">6</span>, <span class="number">4</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">5</span>, <span class="number">10</span>, <span class="number">10</span>, <span class="number">5</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">6</span>, <span class="number">15</span>, <span class="number">20</span>, <span class="number">15</span>, <span class="number">6</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">7</span>, <span class="number">21</span>, <span class="number">35</span>, <span class="number">35</span>, <span class="number">21</span>, <span class="number">7</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">8</span>, <span class="number">28</span>, <span class="number">56</span>, <span class="number">70</span>, <span class="number">56</span>, <span class="number">28</span>, <span class="number">8</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">9</span>, <span class="number">36</span>, <span class="number">84</span>, <span class="number">126</span>, <span class="number">126</span>, <span class="number">84</span>, <span class="number">36</span>, <span class="number">9</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">10</span>, <span class="number">45</span>, <span class="number">120</span>, <span class="number">210</span>, <span class="number">252</span>, <span class="number">210</span>, <span class="number">120</span>, <span class="number">45</span>, <span class="number">10</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">11</span>, <span class="number">55</span>, <span class="number">165</span>, <span class="number">330</span>, <span class="number">462</span>, <span class="number">462</span>, <span class="number">330</span>, <span class="number">165</span>, <span class="number">55</span>, <span class="number">11</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">12</span>, <span class="number">66</span>, <span class="number">220</span>, <span class="number">495</span>, <span class="number">792</span>, <span class="number">924</span>, <span class="number">792</span>, <span class="number">495</span>, <span class="number">220</span>, <span class="number">66</span>, <span class="number">12</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">13</span>, <span class="number">78</span>, <span class="number">286</span>, <span class="number">715</span>, <span class="number">1287</span>, <span class="number">1716</span>, <span class="number">1716</span>, <span class="number">1287</span>, <span class="number">715</span>, <span class="number">286</span>, <span class="number">78</span>, <span class="number">13</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">14</span>, <span class="number">91</span>, <span class="number">364</span>, <span class="number">1001</span>, <span class="number">2002</span>, <span class="number">3003</span>, <span class="number">3432</span>, <span class="number">3003</span>, <span class="number">2002</span>, <span class="number">1001</span>, <span class="number">364</span>, <span class="number">91</span>, <span class="number">14</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">15</span>, <span class="number">105</span>, <span class="number">455</span>, <span class="number">1365</span>, <span class="number">3003</span>, <span class="number">5005</span>, <span class="number">6435</span>, <span class="number">6435</span>, <span class="number">5005</span>, <span class="number">3003</span>, <span class="number">1365</span>, <span class="number">455</span>, <span class="number">105</span>, <span class="number">15</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">16</span>, <span class="number">120</span>, <span class="number">560</span>, <span class="number">1820</span>, <span class="number">4368</span>, <span class="number">8008</span>, <span class="number">11440</span>, <span class="number">12870</span>, <span class="number">11440</span>, <span class="number">8008</span>, <span class="number">4368</span>, <span class="number">1820</span>, <span class="number">560</span>, <span class="number">120</span>, <span class="number">16</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">17</span>, <span class="number">136</span>, <span class="number">680</span>, <span class="number">2380</span>, <span class="number">6188</span>, <span class="number">12376</span>, <span class="number">19448</span>, <span class="number">24310</span>, <span class="number">24310</span>, <span class="number">19448</span>, <span class="number">12376</span>, <span class="number">6188</span>, <span class="number">2380</span>, <span class="number">680</span>, <span class="number">136</span>, <span class="number">17</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">18</span>, <span class="number">153</span>, <span class="number">816</span>, <span class="number">3060</span>, <span class="number">8568</span>, <span class="number">18564</span>, <span class="number">31824</span>, <span class="number">43758</span>, <span class="number">48620</span>, <span class="number">43758</span>, <span class="number">31824</span>, <span class="number">18564</span>, <span class="number">8568</span>, <span class="number">3060</span>, <span class="number">816</span>, <span class="number">153</span>, <span class="number">18</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">19</span>, <span class="number">171</span>, <span class="number">969</span>, <span class="number">3876</span>, <span class="number">11628</span>, <span class="number">27132</span>, <span class="number">50388</span>, <span class="number">75582</span>, <span class="number">92378</span>, <span class="number">92378</span>, <span class="number">75582</span>, <span class="number">50388</span>, <span class="number">27132</span>, <span class="number">11628</span>, <span class="number">3876</span>, <span class="number">969</span>, <span class="number">171</span>, <span class="number">19</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">20</span>, <span class="number">190</span>, <span class="number">1140</span>, <span class="number">4845</span>, <span class="number">15504</span>, <span class="number">38760</span>, <span class="number">77520</span>, <span class="number">125970</span>, <span class="number">167960</span>, <span class="number">184756</span>, <span class="number">167960</span>, <span class="number">125970</span>, <span class="number">77520</span>, <span class="number">38760</span>, <span class="number">15504</span>, <span class="number">4845</span>, <span class="number">1140</span>, <span class="number">190</span>, <span class="number">20</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">21</span>, <span class="number">210</span>, <span class="number">1330</span>, <span class="number">5985</span>, <span class="number">20349</span>, <span class="number">54264</span>, <span class="number">116280</span>, <span class="number">203490</span>, <span class="number">293930</span>, <span class="number">352716</span>, <span class="number">352716</span>, <span class="number">293930</span>, <span class="number">203490</span>, <span class="number">116280</span>, <span class="number">54264</span>, <span class="number">20349</span>, <span class="number">5985</span>, <span class="number">1330</span>, <span class="number">210</span>, <span class="number">21</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">22</span>, <span class="number">231</span>, <span class="number">1540</span>, <span class="number">7315</span>, <span class="number">26334</span>, <span class="number">74613</span>, <span class="number">170544</span>, <span class="number">319770</span>, <span class="number">497420</span>, <span class="number">646646</span>, <span class="number">705432</span>, <span class="number">646646</span>, <span class="number">497420</span>, <span class="number">319770</span>, <span class="number">170544</span>, <span class="number">74613</span>, <span class="number">26334</span>, <span class="number">7315</span>, <span class="number">1540</span>, <span class="number">231</span>, <span class="number">22</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">23</span>, <span class="number">253</span>, <span class="number">1771</span>, <span class="number">8855</span>, <span class="number">33649</span>, <span class="number">100947</span>, <span class="number">245157</span>, <span class="number">490314</span>, <span class="number">817190</span>, <span class="number">1144066</span>, <span class="number">1352078</span>, <span class="number">1352078</span>, <span class="number">1144066</span>, <span class="number">817190</span>, <span class="number">490314</span>, <span class="number">245157</span>, <span class="number">100947</span>, <span class="number">33649</span>, <span class="number">8855</span>, <span class="number">1771</span>, <span class="number">253</span>, <span class="number">23</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">24</span>, <span class="number">276</span>, <span class="number">2024</span>, <span class="number">10626</span>, <span class="number">42504</span>, <span class="number">134596</span>, <span class="number">346104</span>, <span class="number">735471</span>, <span class="number">1307504</span>, <span class="number">1961256</span>, <span class="number">2496144</span>, <span class="number">2704156</span>, <span class="number">2496144</span>, <span class="number">1961256</span>, <span class="number">1307504</span>, <span class="number">735471</span>, <span class="number">346104</span>, <span class="number">134596</span>, <span class="number">42504</span>, <span class="number">10626</span>, <span class="number">2024</span>, <span class="number">276</span>, <span class="number">24</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">25</span>, <span class="number">300</span>, <span class="number">2300</span>, <span class="number">12650</span>, <span class="number">53130</span>, <span class="number">177100</span>, <span class="number">480700</span>, <span class="number">1081575</span>, <span class="number">2042975</span>, <span class="number">3268760</span>, <span class="number">4457400</span>, <span class="number">5200300</span>, <span class="number">5200300</span>, <span class="number">4457400</span>, <span class="number">3268760</span>, <span class="number">2042975</span>, <span class="number">1081575</span>, <span class="number">480700</span>, <span class="number">177100</span>, <span class="number">53130</span>, <span class="number">12650</span>, <span class="number">2300</span>, <span class="number">300</span>, <span class="number">25</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">26</span>, <span class="number">325</span>, <span class="number">2600</span>, <span class="number">14950</span>, <span class="number">65780</span>, <span class="number">230230</span>, <span class="number">657800</span>, <span class="number">1562275</span>, <span class="number">3124550</span>, <span class="number">5311735</span>, <span class="number">7726160</span>, <span class="number">9657700</span>, <span class="number">10400600</span>, <span class="number">9657700</span>, <span class="number">7726160</span>, <span class="number">5311735</span>, <span class="number">3124550</span>, <span class="number">1562275</span>, <span class="number">657800</span>, <span class="number">230230</span>, <span class="number">65780</span>, <span class="number">14950</span>, <span class="number">2600</span>, <span class="number">325</span>, <span class="number">26</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">27</span>, <span class="number">351</span>, <span class="number">2925</span>, <span class="number">17550</span>, <span class="number">80730</span>, <span class="number">296010</span>, <span class="number">888030</span>, <span class="number">2220075</span>, <span class="number">4686825</span>, <span class="number">8436285</span>, <span class="number">13037895</span>, <span class="number">17383860</span>, <span class="number">20058300</span>, <span class="number">20058300</span>, <span class="number">17383860</span>, <span class="number">13037895</span>, <span class="number">8436285</span>, <span class="number">4686825</span>, <span class="number">2220075</span>, <span class="number">888030</span>, <span class="number">296010</span>, <span class="number">80730</span>, <span class="number">17550</span>, <span class="number">2925</span>, <span class="number">351</span>, <span class="number">27</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">28</span>, <span class="number">378</span>, <span class="number">3276</span>, <span class="number">20475</span>, <span class="number">98280</span>, <span class="number">376740</span>, <span class="number">1184040</span>, <span class="number">3108105</span>, <span class="number">6906900</span>, <span class="number">13123110</span>, <span class="number">21474180</span>, <span class="number">30421755</span>, <span class="number">37442160</span>, <span class="number">40116600</span>, <span class="number">37442160</span>, <span class="number">30421755</span>, <span class="number">21474180</span>, <span class="number">13123110</span>, <span class="number">6906900</span>, <span class="number">3108105</span>, <span class="number">1184040</span>, <span class="number">376740</span>, <span class="number">98280</span>, <span class="number">20475</span>, <span class="number">3276</span>, <span class="number">378</span>, <span class="number">28</span>, <span class="number">1</span>&#125;,</span><br><span class="line">                &#123;<span class="number">1</span>, <span class="number">29</span>, <span class="number">406</span>, <span class="number">3654</span>, <span class="number">23751</span>, <span class="number">118755</span>, <span class="number">475020</span>, <span class="number">1560780</span>, <span class="number">4292145</span>, <span class="number">10015005</span>, <span class="number">20030010</span>, <span class="number">34597290</span>, <span class="number">51895935</span>, <span class="number">67863915</span>, <span class="number">77558760</span>, <span class="number">77558760</span>, <span class="number">67863915</span>, <span class="number">51895935</span>, <span class="number">34597290</span>, <span class="number">20030010</span>, <span class="number">10015005</span>, <span class="number">4292145</span>, <span class="number">1560780</span>, <span class="number">475020</span>, <span class="number">118755</span>, <span class="number">23751</span>, <span class="number">3654</span>, <span class="number">406</span>, <span class="number">29</span>, <span class="number">1</span>&#125;&#125;;</span><br><span class="line"></span><br><span class="line">        List&lt;List&lt;Integer&gt;&gt; list = <span class="keyword">new</span> ArrayList&lt;&gt;();</span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i &lt; a.length; i++) &#123;</span><br><span class="line">            list.add((List&lt;Integer&gt;)Arrays.asList(a[i]));</span><br><span class="line">        &#125;</span><br><span class="line">        list = list.subList(<span class="number">0</span>,numRows);</span><br><span class="line">        <span class="keyword">return</span> list;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>






      
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